Hello!
The maximum distance is reached when the angle of elevation is 45°. Denote the initial speed in question as and consider the projection of motion on the horizontal and vertical axes.
Along the horizontal axis the speed is constant (the only force is the gravity one and it is vertical). The initial horizontal speed is Therefore the time of a flight is
Along the vertical axes acts the gravity force which gives the acceleration g downwards. The height is
And at the time the height is zero, i.e.
or
or
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